Mostra el registre d'ítem simple
Rational periodic sequences for the Lyness recurrence
dc.contributor.author | Gasull Embid, Armengol |
dc.contributor.author | Mañosa Fernández, Víctor |
dc.contributor.author | Xarles Ribas, Xavier |
dc.contributor.other | Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III |
dc.date.accessioned | 2010-05-06T12:34:53Z |
dc.date.available | 2010-05-06T12:34:53Z |
dc.date.issued | 2010-04-30 |
dc.identifier.uri | http://hdl.handle.net/2117/7135 |
dc.description.abstract | Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves. |
dc.format.extent | 22 p. |
dc.language.iso | eng |
dc.relation.ispartofseries | arXiv:1004.5511v1 [math.DS] |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en diferències |
dc.subject.lcsh | Recurrent sequences (Mathematics) |
dc.subject.other | Lyness difference equations |
dc.subject.other | Rational points over elliptic curves |
dc.subject.other | Periodic points |
dc.subject.other | Universal family of elliptic curves |
dc.title | Rational periodic sequences for the Lyness recurrence |
dc.type | Other |
dc.subject.lemac | Seqüències recurrents |
dc.contributor.group | Universitat Politècnica de Catalunya. CoDAlab - Control, Modelització, Identificació i Aplicacions |
dc.subject.ams | Classificació AMS::39 Difference and functional equations::39A Difference equations |
dc.subject.ams | Classificació AMS::14 Algebraic geometry::14H Curves |
dc.relation.publisherversion | http://arxiv.org/abs/1004.5511 |
dc.rights.access | Open Access |
local.identifier.drac | 2316532 |
dc.description.version | Preprint |
local.personalitzacitacio | true |